Comparing Length of Tangent and Radius of Circle

Consider a circle with center O and a radius of r. A point A is located on the circumference of the circle. A tangent line is drawn from point A to a point B outside the circle. The distance from point O to point B is represented as d. Therefore, the distance from O to A is $r$, which is the radius of the circle.

Also, segment OB creates a right triangle OAB, where OA is perpendicular to AB. According to the Pythagorean theorem, we can express the relationship among these segments. Specifically, we have:

$$AB^2 + OA^2 = OB^2$$

Given that:

- $OA = r$

- $OB = d$

We can rearrange the equation:

$$AB^2 = d^2 - r^2$$

Now, consider the following quantities:

• Quantity A: The length of the tangent segment AB.
• Quantity B: The radius of the circle, r.

Determine the relationship between Quantity A and Quantity B.